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Properties

Label 16.4.83855159092...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 41^{5}\cdot 172181^{2}$
Root discriminant $48.16$
Ramified primes $5, 41, 172181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-739, -2162, 5946, 4032, -8494, -6076, 809, 3961, 997, -999, -411, 74, 81, 12, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 + 12*x^13 + 81*x^12 + 74*x^11 - 411*x^10 - 999*x^9 + 997*x^8 + 3961*x^7 + 809*x^6 - 6076*x^5 - 8494*x^4 + 4032*x^3 + 5946*x^2 - 2162*x - 739)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 + 12*x^13 + 81*x^12 + 74*x^11 - 411*x^10 - 999*x^9 + 997*x^8 + 3961*x^7 + 809*x^6 - 6076*x^5 - 8494*x^4 + 4032*x^3 + 5946*x^2 - 2162*x - 739, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} + 12 x^{13} + 81 x^{12} + 74 x^{11} - 411 x^{10} - 999 x^{9} + 997 x^{8} + 3961 x^{7} + 809 x^{6} - 6076 x^{5} - 8494 x^{4} + 4032 x^{3} + 5946 x^{2} - 2162 x - 739 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(838551590929703359619140625=5^{12}\cdot 41^{5}\cdot 172181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.16$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 41, 172181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{35} a^{14} - \frac{1}{35} a^{13} - \frac{2}{35} a^{12} + \frac{2}{35} a^{11} - \frac{3}{35} a^{10} + \frac{3}{35} a^{9} + \frac{2}{35} a^{8} - \frac{1}{5} a^{7} - \frac{16}{35} a^{6} - \frac{2}{7} a^{5} + \frac{13}{35} a^{4} - \frac{11}{35} a^{3} + \frac{6}{35} a^{2} + \frac{9}{35} a - \frac{3}{35}$, $\frac{1}{2917984362412630735394415} a^{15} + \frac{7649935874596427209264}{583596872482526147078883} a^{14} - \frac{35778940776246274405267}{416854908916090105056345} a^{13} - \frac{149663843123458462602133}{2917984362412630735394415} a^{12} + \frac{104467693209473276264}{6833687031411313197645} a^{11} - \frac{158851289292095551337177}{2917984362412630735394415} a^{10} + \frac{180018795066198817063387}{2917984362412630735394415} a^{9} - \frac{249521938049123621081288}{2917984362412630735394415} a^{8} + \frac{93921143436574046070224}{2917984362412630735394415} a^{7} - \frac{2963987266078873694473}{64844096942502905230987} a^{6} - \frac{1092372407081886336321421}{2917984362412630735394415} a^{5} - \frac{20567255949869120483396}{47835809219879192383515} a^{4} - \frac{22660090608780635378462}{46317212101787789450705} a^{3} + \frac{3525680177393144383857}{64844096942502905230987} a^{2} + \frac{18434512839387139728137}{138951636305363368352115} a + \frac{114041964963366514256606}{583596872482526147078883}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7588432.87123 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.8.4522441578125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
172181Data not computed